Entanglement is a quantum correlation between two or more qubits such that the combined state cannot be written as a product of independent qubit states. An entangled state exhibits non-local correlations: measuring one qubit instantly affects the measured outcome probabilities of distant qubits, even if no signal travels between them.
A two-qubit state $|\psi\rangle$ is separable if $|\psi\rangle = |\psi_A\rangle \otimes |\psi_B\rangle$—it factors into independent qubit states. If no such factorization exists, the state is entangled.
The four Bell states are the maximally entangled two-qubit states: $|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$ and three others. Measuring either qubit in the computational basis gives 0 or 1 with equal probability, but the results are perfectly correlated: outcomes always match.
Three or more qubits can be entangled in different ways. The GHZ state $\frac{1}{\sqrt{2}}(|000\rangle + |111\rangle)$ is fully entangled but fragile—loss of one qubit breaks the correlations. The W state $\frac{1}{\sqrt{3}}(|001\rangle + |010\rangle + |100\rangle)$ distributes entanglement more robustly.
Entanglement is the defining resource that separates quantum from classical computation. A system of $n$ unentangled qubits can be simulated classically; entanglement enables exponential speedup for certain problems. Entangling gates like CNOT and CZ create entanglement from separable states.
Measuring one qubit of an entangled state projects the other qubits onto definite subspaces. This is the basis of quantum teleportation and other quantum protocols.